Circle Equation Calculator — Standard & General Form
Enter a circle's center (h, k) and radius r to get the equation in standard form and general form, with a coordinate plot.
- 1
D = −2h
−2 × 2 = -4 - 2
E = −2k
−2 × -1 = 2 - 3
r²
5² = 25 - 4
F = h² + k² − r²
4 + 1 − 25 = -20D, E, F are the coefficients of the general form x² + y² + Dx + Ey + F = 0.
How does this calculator work?
A circle with center (h, k) and radius r: standard form is (x−h)²+(y−k)²=r². General form is x²+y²+Dx+Ey+F=0 where D=−2h, E=−2k, F=h²+k²−r². Enter center and radius; the calculator produces both forms and plots the circle.
Formula
How this is calculated
The standard form of a circle equation, (x − h)² + (y − k)² = r², expresses the definition directly: every point (x, y) on the circle is exactly r units from the center (h, k), by the Pythagorean theorem. Reading off the center and radius from this form is straightforward.
Expanding the squares and collecting terms converts the equation to the general (implicit) form x² + y² + Dx + Ey + F = 0, where D = −2h, E = −2k, and F = h² + k² − r². The general form is what you encounter when a circle equation has been algebraically expanded. To convert back from general to standard form, complete the square separately in x and y, recovering h = −D/2, k = −E/2, and r = √(h² + k² − F).
Both forms describe the same curve. Standard form is preferred in analytic geometry and graphing; general form appears more naturally in algebraic manipulation. The calculator computes both and plots the circle on a coordinate grid for visual confirmation.
Frequently asked questions
Group x terms and y terms, then complete the square in each variable: add (D/2)² to both sides for the x group and (E/2)² for the y group. The left side factors into (x + D/2)² + (y + E/2)², giving h = −D/2, k = −E/2. The right side gives r² = h² + k² − F.
If h² + k² − F < 0 then r² is negative, which has no real solution — the equation is an imaginary circle with no points on the real plane. If h² + k² − F = 0, the "circle" degenerates to the single point (h, k).
When the center is at the origin (h = 0, k = 0), the equation simplifies to x² + y² = r². General form becomes x² + y² + F = 0 with F = −r².
Also known as
TG we-Calculate Editorial Team. (2026). Circle Equation Calculator — Standard & General Form [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/circle-equation-calculator
TG we-Calculate Editorial Team. "Circle Equation Calculator — Standard & General Form." TG we-Calculate. 2026. https://we-calculate.com/calculator/circle-equation-calculator.
TG we-Calculate Editorial Team, "Circle Equation Calculator — Standard & General Form," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/circle-equation-calculator
@misc{wecalculate_circle_equation_calculator, title = {Circle Equation Calculator — Standard & General Form}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/circle-equation-calculator}}, year = {2026}, note = {TG we-Calculate} }
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